The goal of math teaching is that the students understand and retain the procedures and concepts. This understanding is best taught concretely. The students should be immersed in math through small daily doses of quality experiences. Formal lessons should be taught, starting with concrete experiences in the form of hands-on manipulatives. Once they have a solid understanding of a concept, students can then work toward representing that concept in the abstract.
Teaching Math Through Experience
Children learn as they observe and attempt to duplicate what they experience. Creating a math classroom that is rich in quality math experiences is essential to providing a learning experience that allows students to excel. As Nummela and Caine suggest in their book, "Making Connections," math teachers must "orchestrate" the experiences to bring the information "to life in the minds of the students" for real learning to take place. They recommend providing experiences that are "similar in complexity, challenge and creativity to those of creative experts," and being spontaneous enough to break from the prepared plan and take teachable moments when they arise. The more math is immersed in the child's environment, the more likely he is to learn, understand and use the concepts he is taught.
Using Concrete Math Experiences
To develop a higher level of understanding in math, children need to start with the concrete. Ruth Beechick makes this point in her book, "A Biblical Home Education." She states, "All young children naturally learn from concrete experience. We just need to let it happen." By managing their environment to provide real-life, concrete math experiences, we are not only teaching children concepts, we are teaching them to apply math to their everyday lives. Such concrete experiences will vary with the concepts being taught. For example, when learning about three-dimensional shapes, the class could explore outside, looking for concrete samples.
Teaching from Concrete to Abstract
Starting with the concrete, students can then work into the abstract in math. Dr. Ronilue Garrison, former professor of "Teaching Math" at William Jewell College, developed a sequence to help teach this concept. In this sequence, she points out the process that children tend to follow as they develop: concrete, semi-concrete, semi-abstract, abstract. When teaching math to children, the teacher needs to start at the concrete level, allowing children to use real-life examples and hands-on manipulatives to gain familiarity with math procedures. Next, the students can work up to drawing pictures to represent the process, reading pictures that represent the process and, finally, using symbols to represent the process. To apply this to algorithms, for example, the teacher would teach multi-digit addition, starting only with base-10 materials. Once the students understand the procedure for adding, they can draw a representation of the materials and continue to work up to representing the procedure with abstract numerals.
Teaching Math through Immersion
Ruth Beechick describes immersion teaching as "nutrition." She says, "You do this meal-by-meal and snack-by-snack." In other words, the teacher looks for opportunities throughout the day to incorporate real-life math situations. Perhaps she will share her classroom budget with the students and walk them through her thinking when selecting the items she will buy with the money. She might also emphasize the time of each transition throughout the day, having students practice reading the clock and announcing when it is time to switch lessons. In small doses such as this, children learn math because they are "immersed" in math mini-lessons throughout the day.
Continual Math Review
It is important to continue to review the concepts throughout the year. Students may forget procedures that they haven't practiced in a while. Using the immersion method allows teachers to help the students retain all of their concepts throughout the year. Teachers can create a checklist of concepts and mark off those concepts when a teachable moment arises, keeping a goal of reviewing each concept once every few weeks. Through math immersion, the students easily retain their understanding of math concepts.
